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.UNITED   STATES  DEPARTMENT  OF  AGRICULTURE 
BUREAU  OF  AGBK  LL  ECONOMICS 


THE  USE  OF  PUNCHED  CARD  TABU  -  EQUIPMENT 

IN  MULTIPLE  CO:  ION  PFT'BL: 


Collected  and  prepared   for   the   use   of 
Statisticians  of  the   Bureau. 


Bradford  B.    Sknith,    In  Charge 
Machine   Tabulation  and  Computing  Section 


.ington,    D.   C 
October,    1923 . 


-  1  - 


The  use   of  punched  c.r-    tabulating  equipment    is  not  new:      It 
was  used  a  nunber   of  years  ago    in    tne   Bureau  of   the   Census  for 
•ration  of  Correlation  tables.     More  recently  it  hae  been  ui 

It   two   of   the    bureaus  of   the   Lepartment  of  Agriculture    m 
%j.      The  writer   in  no   sense  claims  to    be   the    sole  author 
the  methods  herein   set    forth  except    insofar  as  the  -1   tech- 

.ue    is  devised   to    suit   the  equipment  and  problems  of  tne    Bureau  of 
tltural  Economics, 

Especial  credit   is  due   to  Messrs.   Tolley  and  Esekiel  of  this 
Bureau  for  devising  the   least   square  method    (page    1?  et   sea..  )   of 
apj  relation  problems.      The  writer  also  washes   to  express 

appreciation  to  them  for   going  over   the    following   text,    their 

.    ns  and  helpful  criticism. 

The  coefficient  of  Multiple   Correlation,    R,    is  a  measure  of  _ 
the   degree   of  agreement   between  a   given  series  and  the   estimated 
(generally   forecasted)    values  of  the    same    series,    <<hen  these   esti- 
mated values  are  determined  mathematically  from  two  or  more  difier- 
ent   (independent)    series.*     The   following  pages  describe   the   forms 
i  arithmetic  used   in  computing  the   value  of  R  and   in  deriving  the 
on  equation"  or   forecasting  formula.      A  five   variable    tfour 
independent)   example    is  given. 


•Tne   coefficient   of  Multiple  Correlation    (R)  may  be  defined  as   the 
coefficient   of  correlation   (Pearsonian)   between   the   dependent    van- 

ie  and  corresponding  estimates  thereof,    computed  from  the   inde- 
pendent  variables,    i.    e.    if  X' ,    the   estimated  value   of  X,    equals 

b]A  plus  b?B  and   03C plus  a  constant,    where  A,    B,    and  C,    are 

-  —-mrarkent  varrccle^and   bi,    b2,    03,   are  the  net  regression 

efficients,   B  equals  the  mean  product  of  the  deviations  of  X  ai 
X'    fr  lr  respective  averages  divided  by   the   product  o: 

standard  deviations  of  X  and  X'  .      R   is  also   equal   to   the    r  1 

deviation  of  X'  divided  by  the  standard  deviation  of  X.  Also,  U 
standard  deviation  of  X'  squared  is  equal  to  the  mean  proauct  of 
th-  -tinns  of   X  and  X'    from  their   respective  averages. 


Die  data  *r.  •  coded  so  that  the  variation   I 

is  somewhere   be-  depending  upon  the  d.  ion 

■    e  per:,  .      estigu-  The   smaller  S- 

the   simpler  process  of  performing  the  calculations,    for 

the  whole   success  of  using  Holl-  nth  Machines*   lies  in  the   groupi. 

'•  land,    tod  a  grouping--   i 

duction  of  v  xn  reduced  accuracy.      The  c    .1 

be  accomplished  either   oy  subtraction  or   by  ^s 

ii    the    caries  under  consideration  varies   between   120  and   173,    SUD_ 
~t   120  from  all    items,    ar.u  the  differences   oy  p   thus  re- 

ducing the   variation   I      from  0  to    11.      This  ceding  Of  1  in 

way    |  -ates   the   calculation  of  R.      Coding  oy  division  -y 

reduces  accuracy,    but   subtracting  in  no  way  affects  accuracy. 

On  the   following  pages  are  given   the  cod<.  In  the 

example.      A(    3(    C,    &  D  are   the    independent   variables.      X  is  the  de- 
pendent variable    (the   values  of  which  it   is  desired   to   estimate,    or 
"predict".)     Tne    series  should  be    so  arranged   that   the  dependent 
1  ible   is  placed  in  a  column  to   the  right  of  the  independents.     A: 
.    series  have  been   listed,    cross-add   th  Lu  B,    Ct 

CHFCK         an-  r   each  ODsan  uod   list    the    sum  in  tne    n6he<      -  ^a" 

~~'v  CO]  the   right,      Col.    8   Ua   the    tabulation.      This  Ch<  - 

is  not  essential    to   the    solution   out   is  of  considerable  di-    ir.   cht 
•  ic. 

1   typos  of  pun.  sorting 

1    •  ^s*»   used    In  t  -ureau  are    for  rn 

-. 

**  to    be  made   ..  L,    figure  a  that 

can   b<-  rocess. 


IV  FOR       EXAMPLE 

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-  6  - 


Record  No. 

.)- 

B^V 

'35" 

— i«r- 

l 

- 

0 

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k 

6 

14 

It 

2b 

10 

lj 

2 

27 

10 

k 

C 

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170 

3 

1 

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22 

171 

7 

0 

0 

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5 

23 

(*)  0r'  •  mus  33  and  remainder  divided  by  3 

[3)     "       "  "120   "      "  »    ■  o 

"  "    233    «i         it  11       H    5 

(3)        "          "  "          "         "  ti      n    z 


-  7  - 


After  the  data  has  been  coded  and   lifted  as  on  the  preceding 
page*   the   values  ari     than  punched  on  punch  cards,    the   recor  «.  r 

also   being  punched   for   the   purpose   of  Identification,      There   are 

1  columns  on  a  punch  card.      In  nur  example   the  record  number 
was  punched  in  columns  31-2-3,    A  in  columns  34-3,    B  in  3b-7,    C   in  }6-3', 
D  in  UO-1,    X  in  42-3,   and   3   in  44-p.      One  card   is  used  for   each   line-- 
for  each  observation,    that  is. 

ter   the   cards  have    Deen  punched,    tney   should  be    sunned: 
The   sum  of  A,    of  3,    of  C,    of  D,    of  X  and  of  S  should  be  obtained,   also 
the  number  of  cards   should  bo  counted.      The  results   should  be  recorded 
in   seme   such  form  as  follows: 

Form   1 
Sums  and  Means  of  the   Variables. 


w   Items 

A 

B 

C 

D 

X 

s 

171 

1302 

703 

1149 

o30 

.      774 

4b  10 

Means: 

7.6140 

4.1223 

'    6.7193 

"   3.976b 

4.32b3 

26.9591 

The  use  of  the  Check  Sum  first   becomes  apparent  here:      Evidently 
the   sun  of  the    sums  cf  <.,    E,    C,   D(    &  X,    should  equal  the   Sum  of   S;   which 
is   the   case.      Ir.e    sa~e   is   true  of  the  means   (averages).      This  checks 
the  first  additions  used  in  building  up  the  checic  sum  itself,    it  also 
checks  the  accuracy  of  the   puncing;    and  also   of  the  division  in   securing 
the  averages.      The  values   filled   into   the    form  above  are   for   our   example. 

•It   is   sometimes   feasible   to  do    the  coaing  by  punching  the  orignal  values 
upon  the  punch  cards.      Then   sort   tne   cards  on   the   variable   to    be  coded; 
group  the  arrayed  values  into   the  determined  upon  classes  and  gang  punch 
each  group  in  a  new  column  .vith  the  assigned  class  value --such  as  0,    3,    I  c 
The  check  sum  for   the    individual  record   then  can  be  prepared  by   showing 
each  card   separately   in   the    tabulator,    aduing  across,    and   subsequent!., 
..ng  upon  the  card,    after  which  the   procedure    is  as   given  above. 

y 


- 


The  n  i      to   sort  the  cards  upon  the    :  iable, 

iing  the  IS  giVen   below: 


r. 


No. 


I5i_ 


- 


ill 


(10) 


S-jm 


(12) 


iiii 


• 


iiil 


- 


The  cards  being  MM  tht    first  group   t.. 

into  packs--all    the    c.iris    >f    t  i..        :    I 

the  next  value  of  A   in  the    second  pack  &c .    &c  .  .      List   in  c  lumn 
(l)--Frr  the   value  of  A  in  the    first  pack.      Tabulate   this  p.* 

On  the    first   line   in  coIutji   ( 2)  write    the  number   of  c^ris   in   the 
pack;    in  colUBO    (3)  write   the    sun:  of  the   values  of  A  in  this   first 
pack,    in  column    (7)  write   the    sum  of  the   values  of  B  in  this  iirst 
pack;    in  column    (9)    the    sum  of  the   values  of  C    in   this  pack;    in 
c    lumn   (ll) ,   D,    in  column   (13),    X,    in  column    (15)    S.      Take   the 
second  pack,    list   the   value  of  A  in  this  pack  on  the    second   line 
the  ,    and    list  the   corresponding  sum  values  as   for    the    first 

pack.      Repeat   until  all  packs  have  been   so   treated.     When  this   is 
completed  make   the  extensions  for  columns  6,    S,    10,    12,    1^,    &  l6  as 
follows: 

Multiply  the   values   listed  respectively  in  columns   >i    7,    9, 
11,    13  &  13  On  any   line   each  times  the  value    listed  on  the    same    line 
in  column  1.      List  the  products  so  obtained  in  columns  6,    3,    10,    12, 
&  Id  respectively.      Do   this  for  all   lines. 
-a  columns   2,    b,    8,    10,    12,    Ik  &■  16. 

Take   the   cards  and.   s<--rt   them  again,    this   time   on  the    sec 
variable,    3.      Take  a  second   sheet   (Form  2,    Sh.2);   divide   the   sorted 
cards  into  packs,   according  to   the  values  of   B  and   list   these   values 
:    successive   pac/cs   in  column   1.      Tabulate,    list  and  extend   in  a 
nnfl>r  exactly   si.::  to   that  •    a   cards  were    sorted  or.  x- 

cept  igures  need  appear    Is   tir.«  -   ^  &  6. 


roceed  as  for  A  &  B  on  a  new  she<  .,  Sn  3) . 

No   figures  need  appear   in  the  B  colv  .  j,   b,    ],   4  8. 

Sort  rn  :  .  •    (Form  2;    Sh  4):      No   f igure 1   la   the  a,    B, 

Linns:   Column         t     LQ  Lnc  Loslve. 

Sort  on  X  and  repeat   (Form  2,    Sh  5).      He    figures   in  the  A,    . 

C,    rr  D  columns;      Columns  3  to   12  inclusive. 

The  reason  tnat  an   increasing  numcer   of  columns 
oe  ccai-  ::ake  the  extensions  and   sun  thi  lid  give   fig- 

ures already  computed:      Thus  if  we   sort  on  C  ana  extend  its  values 
•s  D,    adding  the  extensions,   we  arrive  at   the    sane   figure  as   i: 

i  onD  and  extended  its  values  tixes  C.)      In  case  difficulty 
is  encountered    in  making  the   figures  check  to   the   check   ran   --  ex- 
rtsd   later  in  connect)   a  with  Form  3--it  nay  be  advisable  to  ma) 
extensions  here  directed  to   be  omitted,    for   the    sake  of  compar 
help  locate    the   errors.) 

Following  are    the    tabulations  of  the   five    sortings  made    in  per- 
ing  tr.p   above    steps   for   our  example.      Note    that   in  each  case  a 
check  is  ai  1  by  adding  up  column  2.      This   should  add.   to   the   total 

IS  problem  as   shown   by   the   data  on  form    1. 
further  check  may  oe  afforded  by  adding  the   sum  columns  for   each  vari- 
able—columns  j,   7,   9,    11,    13  &I5.     These    should  on  every   sheet  a 
the   same   corresponding  figures  given  on  form   1. 


E. 


§5 


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H    (\)    f\l    (0     r-^t     tO    rH    r-t    r-t 


f^J-  -rt  VD  O  to  CT\-=f  Q 
f^-,-i  CO  r-  r*>  n  ,-h  _r»-  So 

r-t  V>  _-    r>  n  r-  r-i   rH 


O 

.-h 
r-  \ 


»-•    ^H    r-t\ 


H    nh-M    CTNVJ3    r^\  r—  CO    O    rH    r-l 


o  o  eo 


OJ 


r^-vX1  ir\-^  ct^vjd  OJ  o 


r-<   m-rf    OJ  CO  ,-1 


f^l; 


H'l^ 


oj  m  r~-vr>  o  io^f 


ni-w  iah 


o  o  vr>  crs  o  tm.3-  j-  vn  j-  ovo 

r^r^\LP>Oj  O  (T^  cr\  <y\  r*~\  r-t  O"  r»- 
i-t    O   C^-»JD   OJ   OJ   OJ   OJ   r-t  CO 

J- 


j  Oto  no  h  t?i(M  r^vjo 
oj  r— vj3  to  co  oj  o  ^*  r*"\Oj 

r*~\  H    r-l    OJ 


d) 


Oi^WinO  LTvVD  J"  «^0  J"   O  J- 
h^t  ocor—Hcr,  loojoj- 


LPv 

mi 

'.  D 


0 


o  i^-d-  m  o  iov^j  oj  oj  vis  o  oj 

h  OJ  roru  OMO  J-  r^r«^^4^( 


mr^vOj    lAO    CT»r-»VJD^t^J-     rH    ^h' 

— <  r-t  jrt    f  \  >-t  r\ 


r*^jj   to.  9  r-  tO  On  O  OJ 


♦  > 

i 

to 


OJ 

B 
C 
o 


o  mv.o  i\j  un' 

OJ   I 

i-h  OJ   c\nlii  CTv  f 


r  meo  fM»\c>»  r-j-  v\  o 


O  r^vO^jD  O  OU3  Q  H  O 

r-t  VJD    l-t    r-t     t^\r-t     CO    LOVJD 


cnr^xOCJNOJ  Of^-ro 
CO  r-t   r^r-  to  CT\OJ 


H 


o  ojcovj^  ionw  o  ono 

OJ  lOf^  CO  l^-M  r*>crvOJ 

rH   OJ    OJ  VJO    CTN  i-t  J" 


VD  OJ  OJ  ^t  LTNt^ro  cr\o  c 


m 


J- 


m 


■) 
So 


o^MtomH.j  ocooj 


I 


OJ  O  OJ  ro>KQ  O  CO  ir  CT\ 


OJ  VJ^    CTNUD    r-l    r^\  r-l 

OJ  r^\ 


O  nj-  OJ   lO  r^\CO  O  C0  KO 
rH   LP\OJ    r^O    r-4    O    G* 

r-t    r-t    OJ    OJ     fA 


CTv 


5 
o 


r-  r«-\  m  on"  o  rH  co  co1 

OJ  r-t    Oj    r-l    OJ    OJ    OJ 


Onfyj  iONmohim 


-  2t  - 


i 
E 


r:1 


o  C"-  ,  in  r 


a> 


- 


!fi  ©  H  Q   '  •   •:    i      .-<  r-» 


o> 


CO 


• 


in 


*_  in  o  c 

CD  iH  x 

F5       n  m  o  w  <:■ 


* 


— 


^ 


J3  10  Is*   <o  c>  o  — <  C 


o  po 


iflOD  H   ,  . — 1|  O 

I 


o  v  o  eg  to  o 

r-t   -  •  :■ 

HH«rn^rtnHvroH 


0^10(DOO<V(J»OCOC 

r-iintoor-r-^r  t>  co 


0»-<CN3tO%-    IfJiDt^CDOO^ 


-   13  - 

Hie  next   step   is   to   transcribe   f:     I  tx         '  fel    (*    to 

5)  of  Tom  2,    to  another   6heet  o:  fona  shown  on   the  follow: 

;.  (For:-.  .>) . 

On  lii  3)   coluj-n  a  list  thv.  figure  taken  fron  Fc  • 

:h.    1  Col.    6.  L)    lir.c.      ?k .  other  fj  on  li.  -   arc 

taken  frou  th<-     it         ihoet    (Ibxi    2;    Sh.    1)  last  line    colui.xs   ~ ,    10, 

16.      The  fifures   filled  into   forr.  3  apply    to  our  lo,    80 

through  the  various  fori.-.s. 
Th<         •      for   line  B-l  cones  from  For:.,  2,    Sh.    2,    last    line, 
nana      ,  10,  12,   14,  <1  16.      ('H-.is  was   the   sheet  used  when  the   cards 
(    sorted  on  B.  ) 

Qjio  data  for  line   C-l   cones   fron  7c  Sh.    3,    last   line, 

:.-nns  10,    12,    14,  &  16.      (This  was   the   sheet  used  when  the  cards 
were   sor  bed   on  C. ) 

[    .    data  for   line    D-l   cones   fro::  Forr.  2,    Sh.    ■  ,    L  st   line, 
colunns  12,    14,   &  16.      (This  was   the  sheet  used  wh-.  carde 

on  D. ) 

B   iron  F0r:.  2,    Sh.    5,    last   lir    . 
,    '■   16.    (This  was   the    sheet  used  when    the   cards  were    sorted 
have  .  to  go  on  the 

(i.O.:  -c)    on  F0m      • 

obtai:  I  igoree   to  go  on    tl  "2B    o 

3,   we  ■  oapatatj  \o  lata  oi  1.      Ji  of  the  vari- 

able eoiuputi  il- 

tip".  -  pf  B,  of  C  4c. 


-  1U  - 


i 

Form  3 
on  of  Suras 

of  Ejtteni. 

f 

A 

B 

C 

1 

X 

S 

i-1 

A-3 

9913 -^ 

2302.6 

)5.0 

?3b7.3 
27.1 

9077.0 

87~; 
323-3 

-4.0 
3177.9 
-1330 

ul33.0 

3393.2 

2593 

374 

35100.5 
27 

B-l 
B-2 
B-3 

2906. b 

4874.0 

^737-1 

l3o.9 

2719.0 
2303.3 

-54.5 

3100.0 

3191.0 

-91.0 

197  - 
1900b .  1 
73b. 9 

C-l 

C-3 

10099.0 

7720.5 

237i.; 

1*051.0 

-^qO.i 
-515.1 

333b. 0 

5200.7 

135.3 

33-37.0 

3097?. 9 

2»*bl .  1 

D-l 
D-2 

E-3 

4730.0 

27C- .  1 
2081.9 

3094.0 

5077.9 

16.1 

19b9U.O 

18332-1 

1361.9 

X-l 
X-2 

X-3 

9S90.O 

3503.^ 
238o. d 

23?73. c 

208ob.2 
270D.8 

V 


roducts  so  obtained  are  list  ctivttly   in  Columns  A,   B,   C 

on  Forr.  3,    line  A-2. 

Next   the   sum  of   tho  second  Variable,    (3  in  our  example;   or  705) 
is  put   into   the  computing  machine  and  multiplied  successively  by  the 
mean  of  B,   of  C,   of  D  &c.  &e«.      The  products   so  obtained  are  listed 
respectively   in  Columns  3,    C,    D,   >xc.   of   Form  3.    line  B-2. 

Next   the   sum  of    the   third  Variable,    (C    in  our  example;    or  11^9) 
is  put  into  the  comtxiting  machine  and  multiplied  successively  by  the 
mean  of  C,    of  D  <£c.  6cc . .     The  products  so  obtained  are  listed  respec- 
tively in  Tolur.ns  C.    D,   &C.   of  Form  3t    line  C-2. 

In  a  similar  manner  the   computations  are  made  for   the  other 

lir.9s  ending  in  "2",-  Form  3. 

"ote;-  In  practice  it  is  most  convenient  to  prepare  Form  3  on 
a  sheet  of  paper  vtfiich  also  carries  Form  1  at  the  top.  The  figures 
for  marring  the  extension  for  lines  of  designation  ending  in  "2"  are 
then  before  the  operator. 

Every  item  on  a  line  ending  in  "2"  is  nor;  subtracted  from  the 
figure  directly  above  it  on  a  line  ending  in  "1."  (Note:  naturally 
-Id  the  minuend  be  greater  than  the  subtrahend  the  difference  ;7ill 
be  a  negative  value.)  These  differences  are  listed  on  the  lines  of 
designation  ending  in  "3n .  These  lines  are  then  transcribed  to  Form  U. 
The  differences  vhich  have  just  been  secured  are  the  product 
nts  and  squared  standard  deviations  (times  1T)  ;  and  are  the  r.eces- 
•  data  for     ;olution  of  multiple,  and  partial  correlation  coef- 
ficients, or  gross  and  net  regression  coefficients.   The  usual  solution 
be  found  in  Yule:   "Int:     ion  to  Statistical  Method."  The  solu- 
D  given  in  the  following,  however,  is  a  "Least  Square"  method,  fi. 
conceived  of  and  developed  by        :olley  and  Ezekiel  of  the  Bureau 


-  if  -  -1«- 

.  . -ultural  Economic  •  ng  th« 

method   is  published  ay  them  in  the  Journal  of  the  An      St 

for  December,    19?5. 

■n    -    In  case   it   ii  jno-ica   to  ma*e   the  extensions   d. 
ner   than   to  use  .  cards  ana   tabulating  machine s--fre que; 

the  case  when  short   series,    such  as  time    series,   are   oeing 
a  multi -columnar   form  should  be  used.      In  the    six  left-.'.. 
List  th      .    •        i     Luna  headings  would  i<  Bf  C,  Dt  X,  4  8. 

-lining  columns   should  be  headed:      irtAB,   AC,    AD     «X,    A|; 
BD,    EX,    BS,    C5,    CD,    CX,    C3;   D2,   DX,   D3;    X2,    XS.      In  the  A2  t 
lt«   the   squares  of  the  values  in  the  A  Column.      In  the  Ab 
write    the  products  of  the  A  items  times  their  corresponding  B   ii- 

When  all  columns  have   been  extended,   add   them,    li  I 
totals  beio.      jn  their  respective  columns. 

Find  the  means    'averages)   of  the- A,    B,    C,   D,    X,    &.  S  colis- 
hlltiply  the    sum  of   the  A  column  times  each  of  the  means  cf  tne 
^x  C ,   D,    X,    &  S  columns  and  write   the  products  below  the    ra 

AD,    AX,    &  AS  columns  respectively.      Multiply  the    i 
B  column  times  the  means  of  the  B,   C(   D,    X,    &  S  columns  and 
the   products  below  the    sums  of  the  B?]   BC,    BD,    BX,  '  &  BS,    colur-.-- 
a   similar  manner  extend  the   sum  of  the   C  column  times   the  meani 
C,    D,    X,    &  S,   and  inscribe   the  products  in  the   C^(   CD,    CX,    & 

iumns.      Also   the   sums  of  the  D,    &  X  columns.      It   is  not  cece_.~_ 
to  multiply  the    sum  of  the    S  column  times  anything. 

subtract  the    last  values   listed   in  the  A*2  column  anu 
columns  to  the   right  thereof  from  the   figures  just  above    them.      H    tne 
•    -  should  be   greater   than   the    subtrahend   the  different 

f   a  negative  value.      These  differences  are  now  to  b*    trail 
rr«4   to  a  new   sheet  of  the  arrangement  shown  in  form  4.      The    ij 
:ences  in  the   columns  commencing  a  i  th  an    "A"   in   their  desi^n^t; 

ferred   to    the    first    line-of  form  U,    designated  as   line  A-  I 
The  differences  in  the  columns  commencing  with  a   "B"   in  the)  Jo- 

nation   (this  of  course   includes  the  B-?  col.)  are    transferred   t 

I  of  form  The  remaining  differences  are    transfc.  '-*- 

BDner.      Tne   so  arranged  differences  constitute   the  Normal  kqua- 

•    square    solution   for   the  value   of  the  net   re 
B.C.    &  D.    on  X. 


-  17  - 
thil  point  the  use  of  the  Check  Sun   (Col.    S)  as  a   I  ^ 

the        •       to    tr.is  point  may  be    shown:      On  Line  A-l    (Form   !>)    the    sum  of 
the    iters  in  columns  A,    B,    C,   D&X  should  equal  the   figure    in  Coluan 

Line  A-l),    thus   checking  the  extension  and  addition  of  all   t 
figures  used   in  connection  frith  delving  these   values. 

Sum  of  the    following* 


Should  check  to- 
The    Sum  of  the   following: 

Should  check  to: 


Line 

Column 

A-l 

B 

B-l 

B 

ii 

C 

■1 

D 

ii 

X 

B-l 


A-l 

C 

B-l 

c 

C-l 

c 

ii 

D 

ii 

X 

C-l 


Son  of  the   following: 


A-l 

D 

B-l 

D 

C-l 

D 

D-l 

D 

ii 

X 

Should  check  to: 


sum  of  the    following: 


D-l 


A-l 

X 

B-l 

X 

C-l 

X 

D-l 

X 

X_l 

X 

ild         ck  to: 

X-l 

and  also    "5"  for   "1"  in  the  a 
check  may  be   secured        It   is  est 

oefon  ftrriod   to  a    : 


(Form  U) 

NORMAL  EQUATIONS 

D 

C 

E 

X 

a 

PM 

• 

23C 

•1 

J. 5 

5.5 

>9-8 

273- 

7M6.U 

i.9 

-.5 

-91.0 

730.9 

W 

-51 

135.3 

1.1 

2081. 9 

>.l 

1361.9 

> 

2386.6 

2706.3 

SOtOTI   1 

1 

J02.6 

.7.1 

528.5 

-133-5 

259  •  3 

3 

-1.0000 

-.O: 

-.1427 

.0530 

-.1123 

-1.2093 

743.4 

136.9 

-34.5 

-9I.O 

73o.9 

M 

"  -3 

-3-9 

l.o 

-3.1 

-32.8 

5 

748.1 

133.0 

12.9 

-94.1 

7C4.1 

b 

7 

■ 

1.0000 

-.1778 

.1103 

.1258 

-.9412 

2373.5 

-513.1 

135.3 

24bl.l 

8 

-46.9 

13.0 

-37-1 

-397.2 

9 

-23-7 

14. S 

Id. 7 

-12 

10 

2307.9 

-484.3 

114. 9 

1933.7 

n 

-1.0000 

.2098 

-.0498 

.'400 

2051.9 

16.1 

13ol.9 

13 

-7.7 

l?.l 

loi  .- 

-9-2 

-10.4 

73.0 

15 

-101. b 

24.1 

406.8 

1963.4 

I   9 

2C08 .  1 

17 
18 

-1.0000 

-.0229 

-1.0228 

D 

:    .0229 

x       lo.l 

19 

1 

/-.1258 

.0U9i 

.0C4d 

:     .054o 

x     135.3 

7.- 

20 

B:     S 

-.0097 

.0025 

:--1330 

x     -91.0 

12.1 

21 

.1". 

.001  6 

-.0078 

.0013 

:    .1079 

x     259-8 

22 

P.M. 

23 

Sq.   Root 

6.94 

24 

3.uc 

17 

-3.05 

:    250.79 

Squart 

J   root  of 

233b. 0 

(See   Line 

X-3,   Col.jy 

1*6 

R     e 

qua  Is     6 . 

94  -k  U 

S           or 

.142 

On  Form  U     Lines  A-3,    B-3,   C-  ,  ire   th» 

ing  the  normal  I  to  be   solved.      The  :  lu- 

tion  is  given  on  lines  1   to   23.   as  described  below    ' 

On  Line   1  wr  irst  normal  Ion,    i.e.,    copy  Line  A-  s . 

divide   every   item  on  Line    1  by    *  .st   item  of  Line    1,    r-  • 

the  algeoraic    signs  end    list    the    quotients  on  Line    2.      In  our  example, 
we  divide   by  o.      The  algebraic   svm  of  the   items  on  Line   2  Columnu 

B,    C(    Lt    ani   X,    should  equal    the    quotient  appearing  in  Column   S  on 
the   same   line.      This  will  not  always  cneck  to   the   last  digit,    owing  I 
the  dropping  of  places  in   the  division.      Nov.*,    draw  a   line  un-ier    the 
figures  just  written   in.      On  Line    3  t;opy   in   the    second  normal  equation, 
that   is,    copy  Line  B-3.      Now  put   the    figure   on  Line   2,    Column  B   (i.e. 
into  the  multiplying  machine  and  multiply  it   consecutively   Dy 
the   items  en  Line   1   in  Columns  E,   C,   D,    X,   and   S,    listing  the   products 
in   the   respective   columns  on  Line  U.      In  our  example,   we  mult 
by  27.1   by  323.3  by    .133.3   by  239.9  end   by   Zlib.j,    giving  as   quotients 
appearing  on  Line  kt    i.e.,    giving   -.3,    -3-9;    l.o;    -3   1;    ~3- ■ 
Now  add  the    items  on  Line   h   to   th*3    ite.-Ls  immediately  above   on   Line    3, 
giving  Line   3.      Careful  attention  must   be   given   to   tne  algebraic    signs. 
Now,   divide  the   figures  on  Line   3   by  the   first   fie^ure   on  Line    , 
reverse    the  algebraic   signs,    listing  tne    quotients  on  Line   6 .      In  our 
exar  :de   by  The  algebraic    sum  of  irst   four    it^ems 

-;uld  check  to   the    last   item  of  the    line,    70^.1,    in   I 
-■^n.      In   liiCe  manner,    the    ilgebraic   sum  of  the   i  on 


•This  is   the    "Doolittle  Method  Bee   Oscar   S.    Adams   -    "Geodesy   -  Ap- 

■    on  of  f  Least   ~  ->  to  the  Adj. 

tion."    -   I?!").      Special  n  #28,    Geodetic   Survey. 


D  - 

Line   o  should  check  to   the    last    item  or.  Line   u,    or   -.9^12.     Now  copy 

D  the    third  normal  equation  on  Line    7;    i.e.,  -one  C-3.      Put  the 

number    la  1  )3  -mn  on  tne    secona   lias,    -.1-27   lato   tne  mull  ng 

machine  ana  multiply  it  consecutively   cy  the    items   la   the  C,   D,    X,   n 
S  columns  of  Line   1,    listing  the  products  in   the  corresponding  columns 
Line   3,    giving  careful  attention  to   the  algebraic   signs.     Next,   put 
the   item  in  the   C   ^  lumn  of  Line   o,    -.1778,    i-'ito   the  multiplying  machine 
■ad  multiply  it  consecutively  oy   the  C(   D(   X,   and  S  column  figures  on 
Ljne    ),  g   the  products  in   their   respective   columns  on  Line   9, 

giving  careful  attention  to  algebraic  signs.     Now,   add  together   for 
each  column  the   values  in  Lines  7,    6,   and  9,    giving  Line    10.      The  ii" 
items  of  this   line    should  check  to   tne.   luot,    similar   to    the  case    : 
Lines  5  a™i  1.     Divide  each  of   the   items  of  this  line   by  the   first   i* 
in  the    line,    that   is,    divide   by  2507.9,    reverse    the    signs  and    list    the 
quotients  on  II.      In  a  manner    similar   to   Line   o,    the    first   ite-^ 

en   this  line    should  check  to  the   last  when  uaaea  together.     Draw  anotr.cr 
line.      jt\  Line   12,    ".rite   the   fourth  normal  equation,    that  is  copy  Line 
D-3.      Put   into    the  multiplying  machine    the   value   on   Line   2,    Column 

I  multiply  it  consecutively   by   the  D,    X,    and   S  column  values  of  Line 
1,    listing  the  prouuets  on  Line    13  in  their  respective  columns,    giving 

-oful   attention   to    the  algebraic    signs.     Next,    place   the  value   on 
Line  o  column  D  into  the  multiplying  machine ,    .1103,  and  multiply  con- 
sec  .  •  y  by  the  values  on  Line   3,    columns  D,    X,   and  S,    listing  the 

ir   re.  re  columns  on   Line    14,    giving  careful  atten- 

tion to   tne  alg»  Next,    put   tne    figure   on  Line    11,    column 

multiplying  machine  and  multiply  consecutively   oy 
the   values  on  Line    10  columns  i),    X,    ana   S,    luting  the   products   in  tneir 


-  21  - 

respective   cr>luuis  on  Line    15,    giving   ciirtfal   attention   to  a]  nL/ns. 

t,  r  •  I  -         12,    - ; .   !■'•  •  L6i 

-lie  ■*  Lren 

Lc  su    of  the   fii 

.1. 
Lret  ite:.  on   "  le,   19?  '.    .   I 

.        .  list   the  17.  '  i  of 

17  should  chock  t  3      "  -.      1".  1 1      w        for 

s   11,   6,   and  2.     Ue  have  now  finished  the  ■forward*    solution  for   tho 

on8i   •  -  tho 

tho  i    Lutioo   to  r  05    vri- 

-"  1.  -  will  t  Wq  are  now  rcidy  for   V 

bf»c         Lution,  lii.es   1   . 

-  I  to    tK    ".".-"" 
the   ■l£n.      T.i~  valni    i;    the  net  r  ."f  icier.  *  ic 

D  X.      Next   in  Golunn   g      lines  IB   to   ?1,    Inclusive,    list   i      -        rse 
order  the  values   in  colUED  X,    Lircs  JL-3,   B-3,    D-3,    D-3.      U  Xt   [ :■:.   lino   19 
Do  In  c .  write  1  revors J  Ob 

:.-  ^luc  on  Line  G,    colucc  X,    revere:- 

ai.-  .  - .        .-.-..,      ri1      •.  c  h  :.  Xt  rev 

the   si  ■  18,    c  .    -  oltipl; 

"  - 

ict   1  01 

c- 
,  •    .  •  . 

val 


Next,   add  toget.vr    tne   \uiues  on  Line    19  in  columns  C  and  L 

sun  luran  X,    Line   19.      ~  '  -   last   sum  1.    I  ion  coefficient 

of  thf  C   on  X. 

Put  a  coefficient  C  -  on  X   into   I 

multiplying  .ie,   and,    having    1   can     I  or  al£-  it 

times   I  ie    j  t    i .  I  iting  the  product      r.  the 

same    line   in  column  rM.      Then,   multiply  it  by   tne  values   In  --  on 

Lines  nd  A-3,    .-.nting  the  products  on  Lines  20,    21  ana.   2k,    respec- 

tively of  the   same  column.      No.         id   the  values  on  Line   20,   columns  B,    C 

1  C   together   ..riting  tr.e    sum  in  column  X  on  tne    same    line  par- 

ticular regard   to  algebraic    signs.      This   1-i^t    ran  written  on  Line   20  in 
column  X  is  the  net  regression  coefficient  of   the   variable  B  on  X.      Place 
it    in   the  multiplying  machine,   and,    having  a  care    to  algebraic    signs,    mul- 
tiply  it   tirr.es   th°   value    listed  beside    it    in  column   S,   writing   the  proauct 
on   the    same    Line    in  column  FM.      Then,    multiply   it   by   the   values  in  cclu 
B  on  Lines   2  and  A-3,    listing  the   products  respectively  on  Lines   <_' 

-  of  the    same  column.     Now,   add  n  Line   21,   and   ir.  columns 

'.,    B,    C,    and  D,   writing  the    sum  in  Column  X  on   the    same    line.  .urn 

net    re^rrssion  coefficient   of  A  on  X.      Place    it    in  the  iying 

machine  a  . tiply   it   times  the    vulu  ide  it  in  column  S,    hav- 

ing a  l  r  algebraic    signs,  and   libt   tne  product   in  column  FM  on  I 

same    line        ill  altiply   it    times  -lue   on  Line  A-3,    column  A,    list- 

roduct   in   the    same   column  on  Line   2^.      No.'  values  on  Lj 

columns  A,    B,    C,    D  I    toge*  ..ould  e^ual    the   value 

0   Line  A-),    column  X,    which   serves  as  a  check  upon  the  a  1  ?n 

r  net  regression  C  X.      There   is  a  d  .ice 


of  ?  between  the   two  va  .    wur  pie,     A  greater  ...  y  may  be    se- 

the  arithmetic   to  a   greater  number  of  pla^  ..jhout   I 

entire    solution.      It  was  deemed  expedient   to  make   the  example  as  simple  as 
possi  i 

We  have  ao  the  net   regression  coefficients,    ehicfa    ire   es- 

sential to  the  forming  of  the  regression  equation  for  predicting  or  esti- 
azi'  oes  of   S.      To  ascertain  to  how  great  a  degree   these    preaic- 

~ns  cc:  -o   the  actual  values,    it   is  necessary  to  obtain  some  measure 

of  agreement   between  them,    the  predicted  and  the  actual.     This  measure  of 
agreemen*  -  .e  coe:  .t  of  multiple  correlation,   R,   defined  on  Page   1. 

To   secure   this  coefficient   of  multiple   correlation,    add  the   values   in  the 
PM  coluaan,    listing  the   sum  on  Line   2.2..     Next,    secure   the   square   root  of 

.5   sum,    given  on  Line   2}.     Finally,    secure  the    square  root  of  the  value 
listed  on  Line  X-3(    column  X,    listing  this  in  the  PM  column  on  Line   25. 

valur    -        .led   into   *  lue    immediately  above    it   on  Line   23,    gives 

the  coefficient  of  multiple  correlation. 

There  are   certain  aids  and  other   checks   in  the    solution  which  can 
be  app  liea  to  h^lp  in  locating  errors.      The  diagonal  terms  of  the  norma.  1 
equations   (23C2.0,    74^--,    2373-5  £»>  are  always  positive  in  sign.      In  me 
ing  the    s-lution  the   figures   listed   immediately  below  these   figures  (to  be 
added  to  them  in  the  course  of  the   solution)  are  always  negative   in   sign, 
.ppearing  above   the    -1.00000  terms  are  always  positive  /  i  .e. 

1,    2307-         I    -3>)- 

icy  is   increased  if  comparatively   small  diagnonal   terms  are 
3  can  be  controlled  by  controlling  the   origins  1  coding.) 
The  Product  Moreen*  22,    is  al-vays  positive    in   sign. 


UNIVERSITY  OF  FLORIDA 


3  1262  08918  7172 


-    2U   - 


THE  REGRESSION  E- 

The  final  step  in  the  arithmetic  ia  to       tha  "r      ion," 

or       citing"  or  "  equation  as  it  is  vario 

.e  down  t:         ;ion  coefficient  of  A  on  X;  ir. 

\ 

~"3.     P>  alue  write  what  V7as  done   algebraically  in 

thU  \  ill  be  of   the  forr.,   ~_: 15.  .     (See  page 

6,    note  (1)).     Then  furtr  rite    the    subtraction  of   the  average  of 

en  from  form  1,    enclosing  all   in  pa:  •  . . 

) 

r  c  form  so  far  will  look  like  this:   .1079  (A  -  35  _  <,,0  ) 

/      ■»     "  -7  .dIhO     ( 

In  an  exact!  nner  treat  the  B,   C,    D,  &  X  scriec.        (Disregard 

1  be  no  regression  coefficient  for  the  X  B< 
of   the  A,   B,    C,   &  D  expressions  should  be  equated  to 
xpression  as  follows: 

♦   .1079(^=21  -  7.61U)     -    .1330(3-30  -  U.1226)  ♦  .05U6(C;120  -  6.71 


) 


( 


D-23S 


.:'22?(— c^--3- 9766  ) 
°  ) 


(  °  ) 


H  -5263 


This   is   '  1 "      :■       Ion  equation.     It  is  only  nee  on  to 

.  uate  for  X,    involvin.  -,    to  put  .uation 

: ul  foi 
.1079  A  -    .0998   B   •  .0137  D  ♦  9^-0552 

ia   is    •  definition 

of  R   in   the  note  on  par.o  1* 


